The Two-Dimensional Random Walk

Suppose that the ant is not forced to step just along a line, but
can move in four mutually perpendicular directions when walking away
from the lamp post. This type of movement is called a {\it 2-dimensional
random walk}.

For example, an ant is standing in the center of a 11 by 11 grid, as
shown in the figure. Each grid square is the size of one step. The ant
can move one step at a time in one of four directions: north, south,
east, or west. The ant cannot move diagonally or take more than one
step at a time. If the ant walks off the edge of the grid, it cannot
return. Discuss the following questions:

Where do you think the ant will most likely be after 10
steps? Will it still be on the grid?

Where do you think the ant will most likely be after 100 steps?
Will it still be on the grid?

Is there a relationship between random walks and coin flipping?
If so, what is this relationship?

Place a checker in the center of a checkerboard as in the figure. Flip
a four sided die labeled north, south, east and west and move the
``ant'' accordingly. After 10 steps, mark on a copy of the checkerboard
the final position of the random walker. Start again from the center
and repeat the same 10-step procedure ten times. Measure the distance
from the origin for each random walker after 10 steps and take the
average of all the distances. If several groups are doing the same
activity, average your averages. What is your result? Compute the
square of the distance from the origin for all walkers and take the
average. What is your result?

In this chapter we have used a very simple model: an ant wandering
back and forth with steps of equal length taken at equal time intervals.
Yet this simple model describes many processes in the real world. How
can this be, since our model is so simple? Very similar results are
predicted by more complicated models that add more randomness: steps of
random length, steps in random directions, steps that take place
randomly in time. It turns out that the predictions of these more
complicated models are similar to ours as long as our model reflects the
average step length, average time between steps, and
average distance from the starting point. Often in science a
simple, easily understood model makes good predictions about the more
complicated real world.

We can do the same simulation using the computer. You will notice
that we graph the mean squared displacement as a function of the
number of steps as we did in the 1-D Random Walk Applet.

Now that you've had a chance to experiment, can you answer these
questions (they are the same questions asked in the 1-D model).

Does changing the number of steps the walker takes effect the
relation between "mean squared distance" and "step number"? Does it
change the "average distance" from the origin?

How does the "mean squared distance" from this simulation
compare to the 1-D case? Are they "the same", "somewhat the same", or
"completely different"? Explain why they cannot be EXACTLY the
same.

Can you think of examples from nature where particles may
move around in a random way?

This lesson is taken from Fractals in Science. Page was
developed by Paul Trunfio and the JAVA applet was
written by Gary McGath.
Please send comments to trunfio@bu.edu. Copyright 1996-2000,
Center for Polymer Studies.